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(i) Show that for any RV's X₁, X2, ···, Xn, n Var (X₁ + X₂ + + Xn) = Cov(Xį, Xj). ΣΣ i=1 j=1 n (Hint: use Var X = Cov(X, X) and expand using linearity in both "slots".) (ii) Show that if X₁, X₂, …, X₂ are independent, Var(X₁ + X₂ + · + Xn) = Var(X₁) + Var(X₂) + • + Var(Xn). (Hint: recall that Cov(X, Y) = 0 if X, Y are independent.) (iii) If X₁, X2, · · , Xn are also identically distributed as X, show that⁹ Var X = ¹ Var X. X, n where X is the sample mean as a random variable 1 X = -— (X₁ + X₂ + ··· + Xn). n

(i) Show that for any RV's X₁, X2, ···, Xn, n Var (X₁ + X₂ + + Xn) = Cov(Xį, Xj). ΣΣ i=1 j=1 n (Hint: use Var X = Cov(X, X) and expand using linearity in both "slots".) (ii) Show that if X₁, X₂, …, X₂ are independent, Var(X₁ + X₂ + · + Xn) = Var(X₁) + Var(X₂) + • + Var(Xn). (Hint: recall that Cov(X, Y) = 0 if X, Y are independent.) (iii) If X₁, X2, · · , Xn are also identically distributed as X, show that⁹ Var X = ¹ Var X. X, n where X is the sample mean as a random variable 1 X = -— (X₁ + X₂ + ··· + Xn). n

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 78E
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Recall the covariance of two RV's X and Y is defined by Cov(X, Y): E[(X – E X)(Y — EY)] = E[XY] – (EX)(EY). := - It has the following properties6: • Cov(X, X) = Var(X). ● Symmetry: Cov(X,Y)= Cov(Y, X). Linearity in the "first slot": for any real numbers a, b, Cov(aX+b, Y) = a Cov(X,Y) and for any RV's X₁, X2, Cov(X₁ + X₂, Y) = Cov(X₁, Y) + Cov(X₂, Y). ● Linearity in the "second slot": for any real numbers a, b, Cov(X, aY + b) = a Cov(X,Y) and for any RV's Y₁, Y2, Cov(X, Y₁+Y₂) = Cov(X, Y₁) + Cov(X, Y₂).
Transcribed Image Text:Recall the covariance of two RV's X and Y is defined by Cov(X, Y): E[(X – E X)(Y — EY)] = E[XY] – (EX)(EY). := - It has the following properties6: • Cov(X, X) = Var(X). ● Symmetry: Cov(X,Y)= Cov(Y, X). Linearity in the "first slot": for any real numbers a, b, Cov(aX+b, Y) = a Cov(X,Y) and for any RV's X₁, X2, Cov(X₁ + X₂, Y) = Cov(X₁, Y) + Cov(X₂, Y). ● Linearity in the "second slot": for any real numbers a, b, Cov(X, aY + b) = a Cov(X,Y) and for any RV's Y₁, Y2, Cov(X, Y₁+Y₂) = Cov(X, Y₁) + Cov(X, Y₂).
(i) Show that for any RV's X₁, X2,..., Xn, n Var(X₁ + X2 + ··· + Xn) = ➤➤Cov (X₁, Xj). i=1_j=1 η (Hint: use Var X = Cov(X, X) and expand using linearity in both "slots".) (ii) Show that if X₁, X2,..., Xn are independent, Var(X₁ + X₂ + ... + Xn) = Var(X₁) + Var(X₂) + ... + Var(X₂). (Hint: recall that Cov(X, Y) = 0 if X, Y are independent.) (iii) If X₁, X₂,..., Xn are also identically distributed as X, show that 1 Var X = = Var X, η where X is the sample mean as a random variable 1 X = ¹² ( X₁ + X₂ + ... + Xn). n
Transcribed Image Text:(i) Show that for any RV's X₁, X2,..., Xn, n Var(X₁ + X2 + ··· + Xn) = ➤➤Cov (X₁, Xj). i=1_j=1 η (Hint: use Var X = Cov(X, X) and expand using linearity in both "slots".) (ii) Show that if X₁, X2,..., Xn are independent, Var(X₁ + X₂ + ... + Xn) = Var(X₁) + Var(X₂) + ... + Var(X₂). (Hint: recall that Cov(X, Y) = 0 if X, Y are independent.) (iii) If X₁, X₂,..., Xn are also identically distributed as X, show that 1 Var X = = Var X, η where X is the sample mean as a random variable 1 X = ¹² ( X₁ + X₂ + ... + Xn). n
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